On weakly (m, n)−closed δ−primary ideals of commutative rings (original) (raw)

On Weakly δ-Semiprimary Ideals of Commutative Rings

Algebra Colloquium, 2018

Let R be a commutative ring with 1 = 0. We recall that a proper ideal I of R is called a semiprimary ideal of R if whenever a, b ∈ R and ab ∈ I, then a ∈ √ I or b ∈ √ I. We say I is a weakly semiprimary ideal of R if whenever a, b ∈ R and 0 = ab ∈ I, then a ∈ √ I or b ∈ √ I. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let I(R) be the set of all ideals of R and let δ : I(R) → I(R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, then L ⊆ δ(L) and δ(J) ⊆ δ(I). Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., I = R) is called a (δ-semiprimary) weakly δ-semiprimary ideal of R if (ab ∈ I) 0 = ab ∈ I implies a ∈ δ(I) or b ∈ δ(I). For example, let δ : I(R) → I(R) such that δ(I) = √ I. Then δ is an expansion function of ideals of R and hence a proper ideal I of R is a (δ-semiprimary) weakly δsemiprimary ideal of R if and only if I is a (semiprimary) weakly semiprimary ideal of R. A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given. 1991 Mathematics Subject Classification. Primary 13A05, 13F05. Key words and phrases. semiprimary ideal, weakly semiprimary ideal, weakly prime ideal, weakly primary ideal, δ-primary ideal, δ-2-absorbing ideal.

On weakly 2-absorbing δ-primary ideals of commutative rings

Georgian Mathematical Journal, 2018

Let R be a commutative ring with {1\neq 0}. We recall that a proper ideal I of R is called a weakly 2-absorbing primary ideal of R if whenever {a,b,c\in R} and {0\not=abc\in I}, then {ab\in I} or {ac\in\sqrt{I}} or {bc\in\sqrt{I}}. In this paper, we introduce a new class of ideals that is closely related to the class of weakly 2-absorbing primary ideals. Let {I(R)} be the set of all ideals of R and let {\delta:I(R)\rightarrow I(R)} be a function. Then δ is called an expansion function of ideals of R if whenever {L,I,J} are ideals of R with {J\subseteq I}, then {L\subseteq\delta(L)} and {\delta(J)\subseteq\delta(I)}. Let δ be an expansion function of ideals of R. Then a proper ideal I of R (i.e., {I\not=R}) is called a weakly 2-absorbing δ-primary ideal if {0\not=abc\in I} implies {ab\in I} or {ac\in\delta(I)} or {bc\in\delta(I)}. For example, let {\delta:I(R)\rightarrow I(R)} such that {\delta(I)=\sqrt{I}}. Then δ is an expansion function of ideals of R, and hence a proper ideal I o...

On (m,n)-closed ideals of commutative rings

Journal of Algebra and Its Applications, 2017

Let [Formula: see text] be a commutative ring with [Formula: see text], and let [Formula: see text] be a proper ideal of [Formula: see text]. Recall that [Formula: see text] is an [Formula: see text]-absorbing ideal if whenever [Formula: see text] for [Formula: see text], then there are [Formula: see text] of the [Formula: see text]’s whose product is in [Formula: see text]. We define [Formula: see text] to be a semi-[Formula: see text]-absorbing ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. More generally, for positive integers [Formula: see text] and [Formula: see text], we define [Formula: see text] to be an [Formula: see text]-closed ideal if [Formula: see text] for [Formula: see text] implies [Formula: see text]. A number of examples and results on [Formula: see text]-closed ideals are discussed in this paper.

On weakly semiprime ideals of commutative rings

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2016

Let R be a commutative ring with identity 1 = 0 and let I be a proper ideal of R. D. D. Anderson and E. Smith called I weakly prime if a, b ∈ R and 0 = ab ∈ I implies a ∈ I or b ∈ I. In this paper, we define I to be weakly semiprime if a ∈ R and 0 = a 2 ∈ I implies a ∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M 2 = 0 is weakly semiprime. We give examples of weakly semiprime ideals that are neither semiprime nor weakly prime. We show that a weakly semiprime ideal of R that is not semiprime is a nil ideal of R. We show that if I is a weakly semiprime ideal of R that is not semiprime and 2 is not a zero-divisor of of R, then I 2 = {0} (and hence i 2 = 0 for every i ∈ I). We give an example of a ring R that admits a weakly semiprime ideal I that is not semiprime where i 2 = 0 for some i ∈ I. If R = R 1 × R 2 for some rings R 1 , R 2 , then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Z m that are not semiprime. We show that every proper ideal of R is weakly semiprime if and only if either R is von Neumann regular or R is quasilocal with maximal ideal Nil(R) such that w 2 = 0 for every w ∈ Nil(R). Keywords Primary ideal • Prime ideal • Weakly prime ideal • 2-absorbing ideal • n-absorbing ideal • Semiprime • Weakly semiprime ideal

On weakly semiprime ideals of commutative rings. Beitraege zur Algebra und Geometrie, 57(3), 2016, 589-597. https://link.springer.com/article/10.1007/s13366-016-0283-9

2016

Let R be a commutative ring with identity 1 = 0 and let I be a proper ideal of R. D. D. Anderson and E. Smith called I weakly prime if a, b ∈ R and 0 = ab ∈ I implies a ∈ I or b ∈ I. In this paper, we define I to be weakly semiprime if a ∈ R and 0 = a 2 ∈ I implies a ∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M 2 = 0 is weakly semiprime. We give examples of weakly semiprime ideals that are neither semiprime nor weakly prime. We show that a weakly semiprime ideal of R that is not semiprime is a nil ideal of R. We show that if I is a weakly semiprime ideal of R that is not semiprime and 2 is not a zero-divisor of of R, then I 2 = {0} (and hence i 2 = 0 for every i ∈ I). We give an example of a ring R that admits a weakly semiprime ideal I that is not semiprime where i 2 = 0 for some i ∈ I. If R = R 1 × R 2 for some rings R 1 , R 2 , then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Z m that are not semiprime. We show that every proper ideal of R is weakly semiprime if and only if either R is von Neumann regular or R is quasilocal with maximal ideal Nil(R) such that w 2 = 0 for every w ∈ Nil(R). Keywords Primary ideal · Prime ideal · Weakly prime ideal · 2-absorbing ideal · n-absorbing ideal · Semiprime · Weakly semiprime ideal Mathematics Subject Classification Primary 13A15; Secondary 13F05 · 13G05

On weakly S-prime ideals of commutative rings

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2021

Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

On Weakly Semiprime Ideals in Commutative Rings

2021

In this paper, we study weakly semiprime ideals of a commutative ring with nonzeroidentity. We give some properties of such ideals. Also, we investigate some results of weakly semiprime submodules of a module over a commutative ring RRR with nonzero identity.

Some results on N-pure ideals

Cornell University - arXiv, 2022

In this paper, we consider the N-pure notion. An ideal I of a ring R is said to be N-pure, if for every a ∈ I there exists b ∈ I such that a(1 − b) ∈ N (R), where N(R) is nil radical of R. We provide new characterizations for N-pure ideals. In addition, N-pure ideals of an arbitrary ring are identified. Also, some other properties of N-pure ideals are studied. finally, we prove some results about the endomorphism ring of pure and N-pure ideals.